Short Exact Sequences and Direct Sums .... Bland, Proposition 3.2.7 .... ....

In summary, the conversation is about a book called "Rings and Their Modules" by Paul E. Bland. The person is currently focused on Section 3.2 which discusses Exact Sequences in ModR. They need help understanding the proof of Proposition 3.2.7. The proof involves showing that M2 is isomorphic to M/Ker g, and that Ker g is isomorphic to M1. The person has two questions regarding this proof and has provided links to discussions about an error in Bland's proposition 3.2.7.
  • #1
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I am reading Paul E. Bland's book "Rings and Their Modules" ...

Currently I am focused on Section 3.2 Exact Sequences in [FONT=MathJax_Main]Mod[/FONT][FONT=MathJax_Math]R[/FONT] ... ...

I need some help in order to fully understand the proof of Proposition 3.2.7 ...

Proposition 3.2.7 and its proof read as follows:

https://www.physicsforums.com/attachments/8082
In the above proof we read the following:

"... ... Then \(\displaystyle M_2 \cong M/ \text{ Ker } g \cong N\) and \(\displaystyle \text{ Ker } g = \text{ I am } f \cong M_1\) ... ... My questions regarding the above are as follows:Question 1I understand that \(\displaystyle M_2 \cong M/ \text{ Ker } g\) by the First Isomorphism Theorem for Modules ... ... but why is \(\displaystyle M_2 \cong M/ \text{ Ker } g \cong N\) ... ... ?Question 2


Why, exactly, is \(\displaystyle \text{ Ker } g = \text{ I am } f \cong M_1\) ... ... ?

Help will be much appreciated ...

Peter
 
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  • #2
Hi Peter,

For question 1, this is a particular case of $(A\oplus B)/A\simeq B$, with $A = \ker g$ and $B=N$. To prove that, you can use the second (or third, depending on the book) isomorphism theorem:
$$
\frac{A+B}{A}\simeq\frac{B}{A\cap B}
$$
taking into account the fact that $A\cap B=0$ if the sum is direct.

For question 2, $\mathrm{img}\: f = f(M_1) \simeq M_1$ because $f$ is an injective homomorphism. If you restrict the co-domain to the image, you get a bijection and therefore an isomorphism. In some cases, $M_1$ will actually be a submodule of $M$.
 
  • #3
 

1. What are short exact sequences?

Short exact sequences are sequences of groups or vector spaces that satisfy certain properties, such as exactness and length constraints. They are commonly used in mathematics and physics to describe the relationships between different objects or structures.

2. Can you provide an example of a short exact sequence?

One example of a short exact sequence is the following: 0 → A → B → C → 0, where A, B, and C are groups or vector spaces and the arrows represent group homomorphisms or linear transformations. This sequence is exact if the image of A is equal to the kernel of B, and the image of B is equal to the kernel of C.

3. How do short exact sequences relate to direct sums?

Direct sums are a way to combine multiple groups or vector spaces into a larger structure. Short exact sequences can be thought of as a type of direct sum, where the middle term represents the "sum" of the other two terms, and the exactness properties ensure that the sequence behaves in a well-behaved manner.

4. What is Proposition 3.2.7 in the context of Bland's work?

Proposition 3.2.7 is a specific result or theorem found in Bland's work on short exact sequences and direct sums. It likely provides a new insight or proof related to these concepts and may be used in further mathematical developments.

5. How are short exact sequences and direct sums used in practical applications?

Short exact sequences and direct sums have many practical applications, such as in algebraic topology, representation theory, and algebraic geometry. They are also used in fields such as physics and engineering to describe and analyze systems with multiple components or structures.

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